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The Relationship Between Apparent Magnitudes and Apparent Brightnesses of Different Stars

The magnitude of a celestial object - star, planet, comet etc - is a measure of its brightness. Magnitudes are measured on a logarithmic scale that is used to determine levels of brightness between other stars. The ratio of apparent brightness between two stars is

\[m_2-m_1=-2.5 log_{10} (\frac{b_2}{b_1})\]

. In this equation

\[m_1, \: m_2\]

are the apparent magnitudes of two stars and

\[b_1, \: b_2\]

their apparent brightnesses (if the power output of a star at a distance

\[d\]

is

\[L\]

then

\[b= \frac{L}{4 \pi d^2}\]

). Suppose then that the apparent brightnesses of two stars differs by a magnitude of 1, so that